Theorem funbrfv 3735

Description: The second argument of a binary relation on a function is the function's value.

Hypothesis

Ref	Expression
funbrfv.1	|- B e. V

Assertion

Ref	Expression
funbrfv	|- (Fun F -> (AFB -> (F` A) = B))

Proof of Theorem funbrfv

Step	Hyp	Ref	Expression
1		brrelex 3197	. . . 4 |- ((Rel F /\ AFB) -> A e. V)
2		funrel 3519	. . . 4 |- (Fun F -> Rel F)
3	1, 2	sylan 448	. . 3 |- ((Fun F /\ AFB) -> A e. V)
4		funbrfv.1	. . . 4 |- B e. V
5		breq1 2612	. . . . . . 7 |- (x = A -> (xFy <-> AFy))
6	5	anbi2d 614	. . . . . 6 |- (x = A -> ((Fun F /\ xFy) <-> (Fun F /\ AFy)))
7		fveq2 3709	. . . . . . 7 |- (x = A -> (F` x) = (F` A))
8	7	eqeq1d 1475	. . . . . 6 |- (x = A -> ((F` x) = y <-> (F` A) = y))
9	6, 8	imbi12d 624	. . . . 5 |- (x = A -> (((Fun F /\ xFy) -> (F` x) = y) <-> ((Fun F /\ AFy) -> (F` A) = y)))
10		breq2 2613	. . . . . . 7 |- (y = B -> (AFy <-> AFB))
11	10	anbi2d 614	. . . . . 6 |- (y = B -> ((Fun F /\ AFy) <-> (Fun F /\ AFB)))
12		eqeq2 1476	. . . . . 6 |- (y = B -> ((F` A) = y <-> (F` A) = B))
13	11, 12	imbi12d 624	. . . . 5 |- (y = B -> (((Fun F /\ AFy) -> (F` A) = y) <-> ((Fun F /\ AFB) -> (F` A) = B)))
14		visset 1804	. . . . . . . 8 |- x e. V
15	14	tz6.12-1 3721	. . . . . . 7 |- ((xFy /\ E!y xFy) -> (F` x) = y)
16		funeu 3523	. . . . . . 7 |- ((Fun F /\ xFy) -> E!y xFy)
17	15, 16	sylan2 451	. . . . . 6 |- ((xFy /\ (Fun F /\ xFy)) -> (F` x) = y)
18	17	anabss7 502	. . . . 5 |- ((Fun F /\ xFy) -> (F` x) = y)
19	9, 13, 18	vtocl2g 1841	. . . 4 |- ((A e. V /\ B e. V) -> ((Fun F /\ AFB) -> (F` A) = B))
20	4, 19	mpan2 694	. . 3 |- (A e. V -> ((Fun F /\ AFB) -> (F` A) = B))
21	3, 20	mpcom 49	. 2 |- ((Fun F /\ AFB) -> (F` A) = B)
22	21	ex 373	1 |- (Fun F -> (AFB -> (F` A) = B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E!weu 1373  Vcvv 1802   class class class wbr 2609  Rel wrel 3165  Fun wfun 3166  ` cfv 3172

This theorem is referenced by:  funopfv 3736  fvelima 3749  funiunfv 3851  cbvfo 3870

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188

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